WEAK SOLUTIONS FOR A GAS-LIQUID MODEL RELEVANT FOR DESCRIBING GAS-KICK IN OIL WELLS*
STEINAR EVJEt
Abstract. The purpose of this paper is to establish a local in time existence result for a compressible gas-liquid model. The model is a drift-flux model which is composed of two continuity equations and one mixture momentum equation supplemented with a slip relation in order to take into account the possibility of flows with unequal fluid velocities. The model is highly relevant for modeling of gas kick for oil wells, which in its worst case can lead to blowout scenarios. The mathematical study of such kinds of models is important for the development of simulation tools that can be enipkiyed for increased control of deep-water well operations. The liquid phase is assumed to be inconifiressible whereas the gas is described by a polytropic equation of state. The model is studied in a framework previously used for investigations of the single-phase compressible Navier- Stokes model. New challenges arise due to the appearance of a generalized pressure term that depends on fluid masses as well as gas velocity. The local existence result is obtained by introducing a suitable transformation along the line of the works [S. Evje and K. H. Karlsen, Cominun. Pure Appl. Anal., 8 (2009), pp. 1867-1894, S. Evje, T. Elatten, and H. A. Ei-iis, Nonlinear Anal., 70 (2009), pp. 3864- 3886] in a free boundary setting. This allows us to obtain sufficient pointwise control of the gas and liquid masses;. The estimates are rather delicate as they must be flne enough to control a possible singular beha.vior associated with the pressure law as well as the slip relation. The existence result is obtained under the assumption of a sufficient small time interval combined with suitable assumptions on the regularity of the initial data, the parameters that control, respectively, the behavior of the initial masse» at the boundaries of the flow domain and the decay properties of the viscosity term.
Key wo)pds. two-phase flow, well model, gas-kick, weak solutions, Lagrangian coordinates, free boundary problem
AMS subject classifications. 76T10, 76N10, 65M12, 35L60 DOI. 10.1137/100813932
1. Introduction. The starting point for the investigations of this work is a one-dimensiional two-phase model of the drift-flux type. This model is frequently used to simulate unsteady, compressible flow of liquid and gas in pipes and wells [1, 2, 4, 5, 6, 15, 18, 20, 21, 25, 22, 10]. The model consists of two mass conservation ec|uations corresponding to each of the two phases, gas {g) and liquid (/), and one eciuation for the conservation of the momentum of the mixture. More precisely, it is given in the following form:
g = 0, (1) dt]aipi] + d:,]aipiui] = O, d] + tt;/J/Ui ] + 0^ [Og/9g U^ + ÜlPluj -\- p] = - q
where Umix = cigUg + aiui and £ > 0. The model is supposed under isothermal conditions. The unknowns are: pi,Pg the liquid and gas densities; ai,ag volume fractions of liquid and gas satisfying ag + ai = 1 ; ui,Ug fluid velocities of liquid and gas; p coiTimon pressure for liquid and gas; and q representing external forces like 'Received by the editors November 8, 2010; accepted for publication (in revised form) May 16, 2011; published electronically August 11, 2011.
http://www.siam.org/journals/sima/43-4/81393.html
tDepartnient of Petroleum Engineering, Eaculty of Science and Technology, University of Sta- Vcinger, 4036 Stavanger, Norway (steinar.evje@uis.no). This author's research was supported by A/S Norske Shell.
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gravity and friction. Since the momentum is given only for the mixture, we need an additional closure law, a so-called hydrodynamical closure law, which connects the two-phase velocities. More generally, this law should be able to take into account the different fiow regimes. In addition, we need a thermodynamical equilibrium model which specifies the fiuid properties. More details will be given in section 2. We refer also to [5, 6, 10, 18, 20, 21, 22, 25] for various numerical schemes which have been developed for the study of the drift-fiux model.
Application. Various gas-kick simulators have been developed for the purpose of studying well control aspects during exploratory and development drilling subject to high pressure and temperature bottomhole conditions. Precise predictions of well- bore pressures, liquid/gas volumes as well as flow rates at the top of the well represent central issues. Clearly, the possibility of blowout occurrences needs to be mitigated in order to avoid human casualties, financial losses (production stop and equipment losses), and finally, but not least, environmental damage. We refer to [1] and the references therein for more information pertaining to this subject. In particular, in [1]
the simulations are based on the drift-flux model (1) equipped with density-pressure relations similar to those used in the present work as well as a slip law that is based on the formulation (32). Development of accurate and robust discretization tech- niques for solving the system (1) is naturally related to a good understanding of its mathematical features (long-time behavior, estimates of various quantities, compact- ness, etc.). In particular, it is clearly of interest to obtain existence, stability, and uniqueness results of various versions of the model (1).
Previous results. Few sucli results seem to exist for two-phase gas-liquid models of the form (1). In [8, 9] we studied a simplified version obtained by assuming that fluid velocities are equal, Ug = u; = u, and by neglecting the external forces, i.e., q = 0.
In addition, we neglected certain gas effects by considering a simplified momentum equation where acceleration terms depend solely on the liquid phase. This is motivated by the fact that liquid phase density typically is much higher than gas phase density.
Consequently, we considered a model in the form + dxlagPgu] = 0, (2) dt[aipi] -I- dx[aipiu] = 0,
"^ + d^p = d:,[ed^u], p,£ > 0.
Assuming a polytropic gas law relation p = Gp'^ with 7 > 1 for the gas phase whereas the liquid phase is treated as an incompressible fluid, i.e., pi = Const, we get a pressure law of the form
(3) p{7i,7n)^
where we use the notation n — agPg and 7n = aipi. In particular, we see that there is a possibly singular behavior associated with pressure at transition to pure liquid phase, i.e., a; = 1, which yields 7n = pi and n = 0. In addition, we have the possibility for vacuum as in the single-phase gas model, i.e., that Pg = 0 which implies that a = 0 and p = 0. Different forms for the viscosity function e have been considered. In [8]
we used
(4) e = e{7n) = ^^—^^^^, /? G (0,1/3),
w]:iereas in [9] we considered
nß
(5) e = e{n,m) = - - j - j , ^ € ( 0 , 1 / 3 ) .
More recently, Yao and Zhu [31] also studied the model (2) in a flow regime where the viscosity coefficient e > 0 was assumed to take the form (4). They gave a proof of the global existence and uniqueness of weak solutions when ß is in (0,1] and thereby improved the result of [8]. They also gave an interesting asymptotic behavior result, and obtained the regularity of the solutions by the energy method. The same authors also presented results for the same gas-liquid model (but constant viscosity term) when the masses 'in,'n connected continuously to a vacuum state 'm = n = 0 [32]. In a recent work we have also studied the model (2) where relevant friction and gravity terms have been included [11]. We also note that the model (2), where both fluids were assumed to be compressible and with a constant viscosity coefficient e, was studied in. [7]. A global existence result was obtained for a class of weak solutions for rather gcjneral inii;ial data.
Why use a viscosity term that depends on volume fraction and fluid densities. Viscosity ßm for a gas-liquid mixture may not be a well-defined quantity just in terms of fluid fractions and single-phase viscosities. The mixture viscosity in fact depends strongly on dynamical processes, including bubble size, flow regime, etc.
Hence, motivated by lab experiments different examples of a viscosity term ßm, where the gas-liquid mixture is considered as a single-phase fluid, have been proposed. Some of them are (see, for example, [26] and the references therein)
(()) ßm ~ ßl (Owen's model), (7) — = — + —^ (McAdams et al.'s model),
ßm ßg ßl
(8) ßm = - j ^ r (Lin et al's model), (9) ßm = yßg + (1 — y)ßi (Cicchitti et al.'s model), (10) ßm = ctgßg + aißi (Dukler et al's model),
(11) ßm = OLgßg + ai{l-\-2.bag)ßi (Beattie and Whalley's model).
Eiere y is defined as mass flux fraction
(12) y = ^ ^ ¡TTo—Ü"'
fbr equal fluid velocities u/ = Ug this corresponds to y = .,J^^^.
The above correlations for the mixture viscosity ßm, obtained from lab exper- iments, reflect that there is room for dependence on both volume fractions ag,ai, densities pi,Pg, as well as fluid velocities ui,Ug. The line we pursue in this work, as in [8, 9], is to consider a choice suggested by the mathematical framework that is employed. However, we now briefly describe why the coefficient used in [8] also seems relevant from a more physical point of view. In that work we studied the model (in Lagrangian coordinates)
9(71 -f {nm)dxU = 0, (13) 9t7n + m^dxU = 0,
fltii -J- r) Pin rn\ — ñ lKlm\f) ?/^ T (^ IC\ ^\
with (14) and
/ 771 N^^+l
^\pi -m)
If we assume that ri < in, then y = ^^^ « i := c (according to the notation used in [8, 9]) for 0 < y < 1. Moreover, typically the liquid viscosity ßi is considerable larger than the gas viscosity ßg-, see (30). Consequently, ßi » ßg and we may approximate as follows by using the viscosity model of McAdams et al. (7):
(16) ^ = ]L + lzJL^^^^,
ßm ßg ßl ßg ßg
Thus, directly motivated by the traditional single-phase viscosity term of the form E = (ßp)^'^^ = Cp^'^^ in Lagrangian coordinates (see, for example, [23, 17, 19, 28, 24, 16, 3, 33]), we may propose a similar viscosity coefficient E = [ßmPm)^^^ for the gas-liquid mixture model (13) where /z,« is a mixture viscosity defined by, e.g., one of the choices (6)-(ll) and Pm is a suitable mixture density. If we define a mixture density pm as
(17) p,n
and combine it with (16), then E = {ßmPm)^^^ corresponds to
:= El -f E2,
where we have used the fact that pg = pi " ^ ; see also (34). Recalling that pi is constant and that c = ^ = c{x) is constant in time, the most "dynamic" part of this viscosity term is the first part,
/3+1
Comparing with (15), we see that E\ coincides with the one that is studied in [8]
except that the coefficient {agßgY^^ has been replaced by a constant.
New results and main challenges. The main novelty of this work, compared to [8, 9, 31, 32, 11], is that the current model allows unequal fiuid velocities, i.e., Ug ^ ui. As a consequence the model, wheii^it is rewritten in terms of Lagrangian variables, contains a generalized pressure term P = P{c, n) — h{ug)g{cn) for appropriate choices of the functions h and g. In particular, the pressure now depends on the gas velocity Ug. More precisely, we consider (1) in a free boundary problem setting where the masses m and n initially occupy only a finite interval [a, 6] C K. That is,
(18) n{x,O) =no{x), m{x,O) = mo{x), Ug{x,O) = Ugfi{x), ui{x,O) = uifi{x)
for X £ [a, 6], and the following boundary conditions are imposed:
(19) r:.{a{t),t) = n{b{t),t)=Q, m{a{t),t) = m{b{t),t) = k*, t>0, where fc* is a constant to be defined later which is related to the slip law. Rewriting the model (1) in terms of Lagrangian variables (details are given in section 2), the free boundary region [a{t), b{t)] is converted into a fixed region [0,1] and the variables {n,m,Ug,ai) are replaced by {c,ri,u), where c = '"~^* and u corresponds to the gas vfilocity. The resulting model takes the form
dtc = 0, (20) dtu + ai^dxu = 0,
dtu + dx[P{c,n) - \u[g{cn)] = dx[E{cn)dxu], x £ (0,1),
where c{x,t) = CQ{X) = " " f f ~'''. The model (20), in view of (18) and (19), is subject to the boundary conditions
(21) c(O,i) = c(l,i) = 0, n{O,t)=n{l,t)=O, í > 0, and the initial conditions
(22) c(a;,O) = co{x), n{x,O) = no{x), u{x,O) = uo{x), x £ [0,1].
N'loreover, we have that
y , g{cn)
k ^ , E ( c n )
^ ^ .*-cn) ^^ ' cn + k* ^ ' {a* - [cn])P+^
H^ere a* < 1 is an upper limit for the amount of gas that can be present, a¡ > 0 is the corresponding lower hmit of liquid, i.e., a* 4- a¡ = 1. Related variables are A;* = pia¡
and a* = pia* appearing in (23). The main challenges we have to deal with are as follows:
• Singular behavior associated with the pressure law (23) similar to the previous works [8, 9, 31].
• Singular behavior associated with the slip law. The gas fraction ag must not he smaller than a critical lower gas volume fraction a*. This is new compared with the previous works [8, 9, 31].
• The appearance of the term [u[g{cn) in (20) due to the use of a more general slip law creates new difficulties. Owing to some technical challenges we have in this work made use of the approximation that u^ « \u[ for small velocities when we derive (20) from (1).
We obtain, an existence result (Theorem 3.1) for the model (20)-(23) for a class of v^eak solutions and for a flow regime where the viscosity coefficient £ in (1) is of the form
(24) s = e{.^) = ^ ^ ^ , ^ G ( 0 , 1 ) .
v/hich in turn leads to the expression E{cn) given in (23). Note that this viscosity coefficient (24) is a natural generalization of (4) studied in [8, 31], which corresponds to the case k* = 0. In fact, if a* = 1, then a¡* = 0 and, consequently, a* = pi and
k* = 0. Then the model (20) reduces to the one studied in [8, 31] with the only difference being that c was defined as c = n/m and not c = in/n.
The main tool in this analysis is the introduction of a suitable variable trans- formation in combination with the continuation method and the pointwise estimate techniques to deal with the singularity of the solution near the free boundary. The transformation allows for the application of ideas and techniques similar to those used in [23, 17, 19, 28, 29, 30, 24, 16, 3, 33] in previous studies of the single-phase Navier-Stokes equations.
Overview. The rest.of this paper is organized as follows. In section 2 we give more details relevant for the model (1). In particular, we derive the Lagrangian variant (20) from (1). In section 3 we state the main theorem with its assumptions.
In section 4 we derive the basic a priori energy estimate Lemma 4.1 and state two lemmas that give pointwise control on, respectively, the masses in, n (Lemma 4.2) and fluid velocity u (Lemma 4.4). In section 5 we derive various lemmas needed for the proof of Lemmas 4.2 and 4.4. Then, in section 6, armed with the results of section 5, we return to these proofs. More estimates are then derived that allows us to make use of standard compactness arguments to prove existence of local (in time) weak solutions.
2. Development of the model. The purpose of this section is to give further details relevant for the drift-flux model (1). Ultimately this will lead us to the simpler model (20).
2.1. Specification of the model (1). To close the system (1), we need to include the following additional equations: The volume fractions are related by (25) ai-^ag = 1.
Thermodynamical laws specify fluid properties such as densities pi, pg and viscosities ßt, ßg. In particular, we will assume that the liquid density has the following form:
(26) pi = pifi H 2 — '
where a/ = 1000 [m/s] is the velocity of sound in the liquid phase and p;,o and pifi are given constants. Here we will assume that pifi = 1000 [kg/ni^] and piß = 1 [bar].
It is often assumed that the liquid is incompressible, i.e., (27) pi = pi,o.
We assume that we consider a polytropic, isentropic ideal gas characterized by
(28) l
In other words, we have (29)
where ag = 316 [m/s] is the velocity of sound in the gas phase. Furthermore, the viscosity for liquid and gas are assumed to be
(30) w = 5 - 1 0 - 2 [Pas], /ig = 5-10-^ [Pas].
Since we have only one momentum equation for the mixture of the two phases, the model must be supplemented with an additional hydrodynamical closure law whose purpose is (;o determine the fiuid velocities ui, Ug through a so-called slip relation. We may assume that the slip relation can be expressed by a general relation
A commonly used slip relation (see, for example, [12, 6, 1]) is given by
(32) f{ag,ui,Ug,pg,pi) =Ug - coUjnix - Cl - 0 ,
where
Umix = aiui.-\-agUg,
and co,ci are flow dependent coefficients, co is the so-called profile parameter (or distribution coefficient) whereas ci is the drift velocity. The gas concentration tends to be highest in the center of the pipe/well for many fiow scenarios, where the local mixture velocity is also fastest. Thus, when integrated across the area of the pipe/well, thse average velocity of the gas tends to be greater than that of the liquid. This effect is represen:ed by the co parameter, ci, on the other hand, represents the buoyancy efFect. Important characteristics of the different fiow patterns can be captured through aj)propriate choices for these two parameters. We refer to the work [6] for numerical e>:amples that illustrate typical transient flow cases with unequal fluid velocities, and we refer to [27] and references therein for more details pertaining to the slip law of the form (32).
For th(j source term q we have two components Q^Ff+ Fg,
where Fg ••- g{aipi + agPg)smd represents the gravity where g is the gravitational constant and 9 is the inclination. Moreover, Ff represents friction forces between the wall and the fluids.
In order to see how pressure p is related to the masses in = aipi and ?i = we observe that the relation (25) can be written as
Using this, we can express the pressure p as a function P of ii and m, i.e., p = P{n,7n).
In particular, assuming that hquid is incompressible, pi = piß, we get from (33) that
which can be plugged into (28) yielding
(35) p{p,)=al
V/e will use this pressure law where we, for simplicity, have set ki = \ for the model we derive in the next section.
2.2. A simplified viscous two-phase modeL As a first step, instead of work- ing directly with the full two-phase model (1) we suggest replacing it by a simpler one. We follow along the lines of [8, 9] and introduce a simplification by replacing the mixture momentum equation by the momentum equation of the liquid phase only.
This is motivated by the fact that the liquid phase density is much higher than for the gas phase, typically to the order of Pg/pi ~ 0.001. The fiquid phase, therefore, plays the dominating role in the mixture momentum conservation equation, as long as the amount of gas does not become too high. We refer to [8] for a careful testing of the validity of this simplification for some typical flow cases. We also neglect external forces like friction and gravity.
To sum up, we consider the model dtn-\- dx[nug] = 0, (36) dtm + d:c[mui\=Q,
dt[mui] -f dxlmuf + P{n, m)] = dx[e{m)dxUmix], Umix = agUg where the pressure law P{n, m) and viscosity term e{n, m) are given by
(37) P{n,in) = (^^—V, e{m) = / " ' ^ - j ^ , , 7 > 1, ß > 0,
\pi—inJ {pi - m)p+' together with the constitutive relations
(38) a ; + a g - l , Ug - CoUmix - Ci = 0, pl = pl,O, Pg = Pg{P),
where Co and ci are assumed to be constants. As will be explained in the following, the slip law Ug - coUmix - Cl = 0 requires that the liquid mass is above a critical lower limit k*, i.e., m. > k*. This information is taken into account in the viscosity coefficient e{in). Similarly, the upper limit for the liquid mass m < pi is also accounted for in the viscosity term (as well as the pressure term) in the same manner as for (4) and (5) employed in previous works. We note that e{in) in (37) is a natural extension of (4)-
We now want to rewrite the model (36) into a form more amenable for analysis.
Our approach is inspired by the work [12]. Given the slip relation
(39) Ug = CoUmix: + Cl,
we introduce a*, a* given by
(40) a* = i, ar-l-a;.
In the following we will assume that
(41) CO > 1,
implying that a* < 1. This is consistent with previous applications of the slip velocity (39) in the context of gas-liquid and liquid-oil flow modeling where Co typically is ranging between 1.0 and 1.5. Moreover, in view of (39) it follows that
, s _ coaiui + Cl _ aiui -\- cia* _ aiui 4-ci(l - a¡) '•' 1 - cottg a* - ag a; - a^*
It is impliciitly assumed that ag < a* (or, equivalently, that a; > a,*) for this slip law to be valid. From (42) we get
(43) am = Ug{ai - a¡) - (1 - Q¡)cx.
N'3xt, we introduce the variable c defined by (44) c =
n n
We assume that the liquid is incompressible: pi — Const, i.e.,
I A^\ m — k*
(4.5) c = ,
71
where k* == piaf is constant. We then apply (43), (44), and (45) and derive the following rtîlations:
(46) m = cn-\-k*,
(47) C7iug = pi{ai - a¡)ug = pi[aiui-i-{I - a¡)ci]
mu \- pi{l — a¡*)ci = inui + d.
where d = p/(l — a*)ci is constant. In other words, mui = cnug — d. Employing (46) and (47) in (36) we arrive at the following form for the system in question:
dtn + da:[nug] = 0, (48) dt{cn]-\-d^icnug] = 0,
dt\cnug] + d We note that
(49) mui - ami "^ '^iPi'^f - Pii^t-i - ^*iWg = Oiipi[uj - u^
Next, we observe in view of (43) that
(50) ai{ui - Ug) = -UgOl - (1 - a*i)ci, (51) ai{ui -{- Ug) = Ug{2ai -a¡) - (1 - a¡)ci.
Combining these two relations we get
(52) af{uf - u¡) = -u¡an2ai - af] -f u¡)
In the folk wing, similar to the work [12], we restrict ourselves to the case that (53) a¡ ^ 0 {a¡ > 0), ci = 0.
In other words, we neglect gas buoyancy effects represented through ci and relevant for vertical flow. This is also consistent with the fact that gravity effects have been n(;glected in the momentum equation in (36), i.e., we consider horizontal flow. Then we have
(54) aipiiuf - u¡) = -piula¡[2 - a^/a] = piu^al [-2 -f ^ ] .
In view of (49) and (54) we get
G{n, m, Ug) := [muj - cnu'^] = pia¡u'^ - 2 4- — 4- pia1[u (55)
I -r uia*"-"^
ai.
* 9 en
noting from (44) that
= —+a¡.
Pi and where we have deflned the function g{-) as (56) g{nc) = fc* cn
cn + fc*
For the pressure law we have
(57) P{n,.n) = ( ^ ^ ) ^ = {^-^ Y = (^^Y := P{c,n),
\pi - mJ \ [pi - fc*] - cnJ \a* - cnJ where a* = pi - k* = pia*g. For the viscosity term e(m) we have
{) 1 ¿ i)
Hence, setting Ug := u, using (39) with the restriction (53) (ci = 0) as well as (55) in the momentum equation of (48), we obtain a gas-liquid model of the following form:
dtn + dx[nu] = 0, ,gg^ dt[cn[ +dx[cnu] = 0,
dt[cnu] + dx[cnu'^] + dx[P{c,n) - u'^g{cn)] = —dx[£{cn)dxu].
Co
In the following we may absorb the constant 1/co into the viscosity term e without loss of any generality.
Lagrangian coordinates. Following the approach of the works [8, 9, 31], which in turn is motivated by studies for the single-phase gas model, we suggest studying the modeL(59), described in terms of the variables (c, n,u), in a free boundary setting:
+ dx[nu] = 0, (60) dt[cn] + dx[cnu] = 0,
dt[cnu] + dx[cnu'^] + dx[P{c,n) - u'^g{cn)] = dx[e{cn)dxu], a{t) <x< b{t).
and Í > 0. Initial data are
(61) n{x,t = 0) = no{x), c{x,t = 0) = co{x) = " ' ° ^ - ' | ^ , u{x,t = 0) = uo{x).
for X £ [ao, bo], where ao = a{t = 0) and i»o = K^ = 0). Boundary conditions are set as follows:
(62) n{a{t),t) = n{b{t), t) = 0, c{a{t), t) = c{b{t),t) = 0.
Here a{t) and b{t) are free boundaries, i.e., the particle path separating the gas-liquid mixture and the vacuum like state corresponding to n = 0 and c = 0, satisfying (63) ^ ^ ^ a ( í ) , í ) , n{o,{t),t)=c{a{t),t)=^,
(64) I = u{b{t), t), n{b{t), t) = c{b{t),t) = 0.
Lc!t us introduce Lagrangian coordinates by using the transformation {x, t) -^ {S,, T) given by
(65) Í = / ' cn{z,t)dz, r = t, Ja(t)
observing that
Mt) çbu
I cn{z,t)dz= I cn{z,t = 0) dz = Const = 1.
J a(t) Jai)
This implies that ]a{t), b{t)] is converted into the fixed interval [0,1] and d d _ d d _ d
dt dx dr dx dí, Applying this transformation in (60) gives
drU + ndxU = 0, (66) dr]cn]-\-,[cn]d:,u = O,
]cn]drU + dx[P{c,n) - u''g{cn)] = dx]£{cn)dxu] in 0 < ^ < 1.
In other words,
drTi + cn^d^u = 0, (67) dr{cn] + ]cn]^d^u = O,
drU + d¿P{c,n) - u'^g{cn)] = d¿e{cn)]cn]d^u] in 0 < ,f < 1.
We now replace (r, ^) by {t,x). Moreover, an easy calculation shows that (67) corre- sponds to
dtc = 0, (68) dtn + cn''d^u = 0,
dtu + dx]P{c, n) - u''g{cn)] = d:,[E{cn)da:u] in 0 < .x < 1, with boundary conditions
(69) c(0,Í) = c(l,Í) = 0, n(0, t) = n{l, t) = 0, í > 0,
and with initial conditions
(70) e{x,O) = co{x), n ( x , O ) = n o ( x ) , u{x,O) = uo{x), X G [ 0 , 1 ] ,
where c{x,t) = CO(,T) = ""„„(J^ ' • Moreover, we have from (57) and (58) that (71)
E{c7i) = e{e7i)[en] = ,^ ^ " _ . ^ „ Finally, in this work we shall make use of the approximation that \u\ » u'-^, which is reasonable for small u, such that we replace v?g{evL) by |u[g((ciz) in the third equation of (68). The motivation for using this approximation is to avoid some technical difficulties not yet solved. Hence, the model (68)-(71) is now consistent with the model (20)-(23).
R e f o r m u l a t i o n . For the analysis of the model (20) it will be convenient to introduce the function Q{e, ri) given by
(72) Q{e,7i) = , which corresponds to 7i — a* .n Q a* — en I + cQ
A similar approach was also used in [8, 9], however, for a different model with equal fiuid velocities. The following nice relation holds for Q{e,7i):
Q{e, 7l)t = QcCt + Qnnt = Qn^
1 en
a* — e7i {a* — c?i)^/ (a* — C7i)^nt =
~ ~7~'* w^a: =^ -a*cQ{c,n)'^Ux.
{a — c?x)^
Hence, the system (20) can be replaced by dte = 0, (73) dtQ + a*cQ'^dxU = 0,
dtU + d^[P{Q) - \u\g{eQ)] = d:c[E{eQ)d:,u] in 0 < ,T < 1, with
P{Q) = Q{e,7i)'', E{ai) = £(cn)[c7x] = [cQ]^+^ := E{eQ), cQ
since
en = a cQcQ'
Clearly, g as a function of cQ possesses the same features as g{e7i). Most importantly, it is always bounded as a function of its argument cQ. This feature of g will be crucial for the analysis that follows in section 4. Boundary conditions for our system (73)-(74) are (in view of (72) and (69))
(75) c(0, t) = c ( l , t) = 0, Q ( 0 , t) = Q{l,t)=O, t> 0.
Initial conditions are (in view of (72) and (70))
(76) c(.T,O) = co(,x), Q{x,O) = Qo{x) = ^ - ^ , u{x,O) = uo{x), , T G [ 0 , 1 ] .
a* - Cono
3. A local existence result. In this section we give the main theorem of this paper, a local existence result for the model (20)-(23). Before we state the main result we describe the notation we apply throughout this paper. W^''^{I) = H^{I) represents the usual Sobolev space defined over / = (0,1) with norm || • |[n/i,2. Moreover, L^'{K,B) v4th norm || • \\LV{K,B) denotes the space of all strongly measurable, pth- power integrable functions from K to B where K typically is a subset of R and B is a Banach space. In addition, let C"[O,1] for a G (0,1) denote the Banach space of functions on [0,1] which are uniformly Holder continuous with exponent a. Similarly, let C"'"/^(.Dr) represent the Banach space of functions on DT = [0,1] x [0, T] which are uniformly Holder continuous with exponent a in x and a/2 in t.
Now we will give a precise description of tlie assumptions on the initial data (co,no,wo) that are required for the existence result to hold. For the analysis that follows in sections 4-6 we need more precise information about how fast 'no is de- creasing towards zero and how fast nio will decrease towards the lower limit k*. We now specify this more precisely as well as information about regularity properties of the initial data (co,7Zo,'Uo). In addition, information about the constraints on the parameters a, ß, and 7 will be given. Here a > 0 is related to the decay towards the boundary points a; = 0,1, /3 > 0 is related to the viscosity term E{cn), and 7 > 1 is relevant for the pressure law P{c,n); see (23).
3.1. Assumptions. The analysis throughout the whole paper depends on a set of assumptions.
(Al) The following assumptions are made regarding upper and lower limits on the inicial masses ino{x),no{x):
(77) Ai(l>{x)" < ino{x) - k* < ^2</'(.x)°, (78) BicP{x)^"/' < iio{x)
for some constants Ai, A2, Bi,B2,a > 0 and 4>{x) = x{l — x). These bounds can then be translated into upper and lower limits for the variable c which is used in our model (20) instead of 777,. In particnlar, this implies that for suitable choices of G\ and C2, for instance, Gi = A1/B2 and G2 = A2/B1, (79) Ci(/>(.x)"/'' < co{x) < G2(p{x)'''\
Clearly, c{x,t) = co{x); hence these estimates hold for c{x,t) for all times t.
(A2) We must ensure that the pressure law P = Q'^ is well defined at initial time.
Fo:r that purpose we here assume that the initial gas phase does not vanish at some point (no transition to pure liquid flow only), i.e.,
(80) a*i < ai,o{x) < l - 6
for some ô > 0. In view of (72) and (78) we may assume that there are positive constants A,B such that (see Remark 3.1)
(81) ^4>{x)'''^' <Qo = Q{co,no) <
Thus, it follows that (82)
(A3) Assumptions on a,ß G (0,1) and 7 > 1 are as follows:
7 > ^ a , 7 > / 9 + ^ , 7 > ^ A - 7 > 1 + ; S - - ,o 3 3 4 a
(A4) Regularity assumptions on the initial data are as follows:
(83) eo,uoeH\[0,l]), [coQo]^ e ^^^[0,1]),
In particular, we note that ^ ¿ ^ G ¿^([0,1]) due to (79) and (81) since /Q <?i>(x)* dx < 00 for s > - 1 .
B.emark 3 . 1 . Note that the first assumption (77) ensures that the initial liquid volume fraction aifi is above the lower critical limit a¡* throughout the whole domain except at the boundary points x = 0,1 where aiß = a^*, i.e., ag^ = a*.
The second equation (78) puts an additional constraint on the pressure behavior at the end points .T = 0,1 by assuming that a vacuum state p = 0 occurs.
Concerning the inequalities (81) we note that a* - aio — pi - mo = pi{l - a/,o), from which it follows from (80) that piS < a* - eno < pia*g. Hence, in light of (72) and (78)
^ 0 ( x ) - / ^ < ^ < Qo = ^ r ^ ^ < ^ < ^ 0 ( x ) 3 " / ^ Pia* pia*g a* - aio piS piS
From this relation we can find constants A and B such that the inequalities (81) hold.
3.2. M a i n r e s u l t . Under the above assumptions, our main result can be stated as follows.
T H E O R E M 3 . 1 . Under the assumptions (Al)-(A4), there exists a positive time constant Ti > 0 (small time) such that the free boundary value problem (20)-(23) admits a weak solution {c,n,u){x,t) on [0,1] x [O,Ti] in the sense that
(A) we have the following regularity:
c , ? z , w G L ( [ 0 , l ] x [ 0 , i?(cn)w, e L ~ ( [ O , l ] X [
Moreover, the following pointwise estimate holds for n:
where ß = ß{B,C2,Ti) > O is a small constant determined in Corvllary 4.3.
(B) Moreover, the following equations hold.
et = 0, nt + en^Ux = 0,
(c, 7i)(x,0) = {eo{x), n(,{x)) for a.e. x G (0,1) and any t>0,
(86) I I [u0t + (P(c,7i) - \u\g{en) - E{en)ux)4)x\ dxdt
uo{x)4){x,O)dx = 0 for any test function (f){x,t) G C¿"([O,1] x [O, Ti)).
4. Basic estimates. Below we derive a priori estimates for (c, n, it) which are
assumed to be smooth solutions of the initial boundary value problem (20)—(23). Then
the method of lines can be used to construct approximate solutions of (20) and derive corresponding estimates.
In the following we will frequently take adyantage of the fact that the model (20) c.an be rev/ritten in the form (73)-(76) which is more amenable for deriying yarious useful estimates.
Some words about the notation used for constants. We shall use G and C¿ (i = 1,...) to denote positiye constants that depend only on the initial data and other known constants as stated in the assumptions given in section 3. Some places we also use D, E, F, K and Di, Ei,Fi, Ki {i = 1,...) for the same purpose. In particular, tJiese constants, G, D, E, F, K, are independent of tjie positive constant M which appears in Lemma 4.4. On the other hand, we use G and C¿ {i = 1 , . . . ) (similarly for D,E,F,K) to represent constants that also, however, include dependence on the positive constant M.
4.1. A priori estimates. Now we derive a priori estimates for {c,n,u) by mak- ing use of the reformulated model (73)-(76) described in terms of {c,Q,u). We start with the standard energy estimate which is slightly more involved in our case compared to a standard single-phase model due to the appearance of the new term [u\g{cQ).
LEMMA 4.1 (energy estimate). We have, under the assumption (A4), for each T > 0 the basic energy estimate
(87) fflu^ + ^ ^ Q{c,n)''-')dx + f i\cQ{c,n)f+\ux)'dxds < G{T),
7Q \ Z a C[-J—ij / JQ JQ
where 0 < t < T and G{T) is a constant that depends only on the regularity of uo and Qo as stated in assumption (A4).
Proof We multiply the third equation of (73) by u and integrate over [0,1] in si)ace. Applying the boundary condition (75), the fact that g{0) = 0 = E{0), and the ecjuation
(88) ^ , ^ ( Q - ^ ) . + Q V . = O,
olitained from the second equation of (73) by multiplying with Q''~^, we get
^''^ Jt Jo (5'"' + ¡ ¡ M 7 ^ ^ " f
W^e may ccinsider the splitting
\u[g{cQ)ux = [ u [ [ c Q ] - ( ^ + i ) / 2 3 ( c Q ) • [cQ]^^+'^I^Ux = a - b .
In light of Cauchy's inequality with e, ab < (l/4e)a^ 4- e6^, we may conclude that (90) / [u[g{cQ)uxdx<{l/Ae) [ u'[cQ]-^^+'^g{cQ)''dx+ e [ [cQf+\ux)''dx.
Jo Jo Jo For the first term on the right-hand side of (90) we observe that
since the function g{y) = y^~^/{k* + {a* + k*)y)'' clearly is bounded for all y > 0 where 0 < ^ < 1. Hence,
,.1 çl ni
(92) / ]u\g{cQ)uxdx<{l/4£)Ci i u''dx + e E{cQ){ux)'^ dx.
Jo Jo Jo
We plug (92) into the right-hand side of (89) and observe that the flrst term on the right-hand side of (92) is handled by the application of Gronwall's lemma. Moreover, by an appropriate choice of e the second term can be adsorbed in the corresponding term appearing on the left-hand side of (89). From this, (87) follows. G
Remark 4.1. Note that the special properties of g{-), it is bounded and goes through zero, play a crucial role in the above proof of the fundamental energy estimate (87).
The next lemma deals with upper and lower estimates for Q{c, n). In view of the fact that n = a* ^ J / Q ; these estimates can be directly translated into corresponding estimates for n. Along the line of [3, 33] we use the continuation method, in com- bination with seniidiscrete versions of the various lemmas derived below, to obtain pointwise control on Q (and n).
LEMMA 4.2. Under the assumptions of Theorem 3.1 there exists a time Ti = Ti{A,B,M, ]]UO]]H^) such that if
(93) iyl<^(.x-)3"/''<Q(c,7z)<2i?#£)3«/'' Vie[O,T],
where T £ (O,Ti] is any fixed positive constant, then we have the following estimate:
(94) ^-Act>{x)'^/''<Q{c,n)<'^B4>{x)'"/" Ví6[0,T].
The proof will be given in section 6 after a set of useful lemmas has been derived.
We also note here that Ti is defined as
(95) T i = m i n { f i , f 2 , f 3 , f 4 , l } ,
where fi is given by (144), f2 by (148), f^ by (154), and í j by (126). We use the
"tilde" notation to indicate that these constants depend on the positive constant M appearing in Lemma 4.4.
COROLLARY 4.3. If Q{c,ri) satisfies the lower and upper bounds of (94), then the following bounds hold for a:
where ß = ß{B, C2,Ti) > 0 is a small positive constant.
Proof. We observe that Q{c,n) is a strictly increasing function with respect to n for a fixed c and its inverse is given by Q~^{c, y) = Û* Y+7~I which also is an increasing function. Consequently,
At the same time it is clear from the upper bound of Q given by (94) and the expression (72) for Q that
cQ=- <C{B,C2,Ti).
a* - cn
In other words, since ^.™^^ tends to infinity as c?i —^ at, en must be a certain distance p > 0 below of a*, and p depends only on B, G2, and Ti, i.e., c?i < a* - p{B,G2,Ti).
Similarly,
^ ( \ ( ' ^ A '^^x)'"^' la*A<f>{x)^^/^
The next lemma concerns the pointwise control of the fluid velocity u. It involves a positive constant M such that M > {1, [|ÍÍO[[OO} and is determined by the inequality (132). The proof of Lemma 4.4 will also be given in section 6.
LEMMA 4.4. Under the assumptions of Theorem 3.1 and (93), if (97) \\u\\,^^<M Vi€[O,T],
then we get
(98) \\U\\L^<\M V Í G [ O , T ] .
5. Proof of some lemmas. Before we return to Lemmas 4.2 and 4.4 and give proofs we shall derive some more regularity estimates. These results will be derived under assumptions (93) and (97). In other words, we assume the following estimates:
(99) ^ > l # c ) ^ " < Qi^^t) < 2 B 0 ( x ) î " Vi G [0,T], (100) \\u{-,t)\\oo<M V Í G [ O , T ] ,
for an appropriate time T to be specified later. The results in this section are obtained for constants C3, C5, and GQ that are independent of M when
(101) i < r
tlowever, the constant G4 appearing in Lemma 5.2 depends on M. Consistent with vdiat jve have said before we shall in the following calculations use the convention that Gi represents a constant that depends on A4, whereas Gi depends only on known constants and constants appearing in the assumptions (Al)-(A4).
LEMMA 5.1 (additional regularity). Under the assumptio7is of Theo7-em 3.1 as well as (99) and (100), we have, for t G [0,T], the estimate
j
(102)
for a constant Gz = GMCOQO\^\\H^A\CO\\H^ AUO\\L^.,B,G,GI,G2,T).
Proof. From the second equation of (73) we derive the equation (103) J_([cQ]^),4-[cQ]^+i„,=0.
jDQj
Using (103) in the third equation of (73) and integrating in time over [0,i] we arrive at
(104) u{x,t)-uo{x)+ / dJp{Q)-\u\g{eQ)) ds = - Jo ^ / pa
Multiplying (104) by ßa*{dx[cQ]^) and integrating over [0,1] in x, we get
/•I
Jo
= ¡\dx[cQf){dx[coQof)dx Jo
- ßa* I {dx[cQf) [{u -uo)+ I dx(P{Q) - \u\g{cQ)^ ds] dx
< £i / {dx\cQ\) dx-\-Ki j (92:[coQo] ) dx Jo Jo
/.I rl
Jo Jo
dxlcQf)'^ dx + K3 I (Í dx (P(Q) - |w|ö(cQ)) ds^ dx.
(105)
where we have used Cauchy's inequality £ > 0, a6 < (l/4£:)a^ -\-eb'^. For the last term we can apply Holder's inequality for the term JQ{P{Q) - \u\g{cQ))xds and estimate as follows;
(106) / ( Í dx(P{Q)-\'u.\g{cQ)^ds^^dx<t j I {dx[P{Q)-\u\g{cQ)\^''dxds Moreover, we have for P{Q) = Q''
dx
(107)
o
cQ''
dx
^c,j dx
<K, / \{[cQf)xYdx-^K, ¡
Jo ^ -' Jo
The first term on the right-hand side of (107) is controlled by the use of Gronwall's lemma, the second by assuming c G H^, i.e., assumption (A4). Here we have also used assumption (Al), which ensures that
and in view of (99) we conclude that c^ ~
for 7 > 1/3. Moreover,
< IU{B,Gi)
02
c
î"^-ï
if 7 > 1/3. which clearly is satisfied. Similarly, we have
/ {[\u\g{eQ)]:,)'dx<2 [ {\u\,)^g{eQ)Ux+ 2 I ]u|V
^0 Jo Jo
= 2 (108)
j
E{eQ){ux)^ dx + K^Ad^ j [{[eQf)x] ''dx.
vdiere we have used that g{cQ)'^[eQ]-^^+'^^ and g'ieQ)'^[cQ]'^^^-'^^ are bounded even independent of the bound (99), in view of the special form of the function g{-). We also note that we have used the estimate (100).
The conclusion of combining (105)-(108), where we also apply (87) for estimating the first term of the right-hand side of (108) as well as the fourth term on the right- hand side of (105), is that
J
j {M^ + M) I [dx [eQf)'' dx) ds.
v/here/^s = K8{B,C,Cu ||[coQo]''||//', ||CO[[ÍÍ>, [[UO|]L2,T) and we have employed (101).
Thus, application of Gronwall's inequality gives the estimate /
Jo
dx < /<r8exp{(l -^ M)i} < C3, where we again have used (101). By this, (102) has been proved. D
LEMMA 5.2 (additional regularity). Under the assumptions of Theorem 3.1, (99), and (100) we have, fort G [0,T], the estimate
(109) /
for constants C4 = Ci{Ci,A,B,AÍ,T) and C5 = C5{Ci,Á,B,T) determined by (113).
Proof Integrating the third equation of (73) over the interval [x, 1] and using the boundary condition (75), we get
/ ut dy - {Q'' - \u\g{eQ)) = -E{eQ)ux.
Jx This corr(3sponds to
. , , Q'' \u\gicQ)
E{eQ)Jx '-''''-E{eQ)-^{^
Consequently, we can estimate as follows:
Hence, integrating over [0,1] gives us (110)
i\ux)'[
Jo
Clearly, the last term on the right-hand side is bounded in view of (100), the properties of g{-), and 0 < /3 < 1. Similarly, the middle term is bounded, in view of assumption
(Al) and (99) since
^_3^_l 47-3/3-1 < ç I g , . .-ia(3/3+l) , . .|a(47-3/3-l) (111) _ 1 vv / -rx J
if
a(37 - 1 - 3^) > 0;
that is, 7 > /S 4- ^. For the first term on the right-hand side of (110) we have by Young's inequality that
[ dyV
for a suitable choice of Kio. For the last choice of Kio = Kio{Gi,A) we have used that
if the following relation holds:
-a{3ß + l) + 2>0, i.e., 2 >a(3/3 4-l).
If 0 < /? < 1, it follows that a must satisfy that 0 < a < 1/2. Thus, from (110)-(112) it follows that
(113) I {uxf[cQf+^dx<Ku{l + M^) + KiiU
where Ku=Kii{Gi, A, B,T). D
LEMMA 5.3 (additional regularity). Under the assumptions of Theorem 3.1, (99), and (100), we have, fort £ [0,T], the estimate
(114) [ {dtu)''dx+ [ I E{cQ){utx)''dxds<Ce Jo Jo Jo
for a constant CQ = CQ,{C, Ci,A, B,T).
Proof We have from (73) that
Ut + {P{Q) - \u]g{cQ)), = {E{cQ)ux)x, which gives
)ux - P{Q) + ]u]g{cQ))
I xt
E{cQ)uxt - P{Q)t)^ + {\u]tg{cQ) + \u]g{cQ)t)^.
A^Iultiplying by Ut, integrating over [0,1], and performing integration by parts, where we use the boundary condition (75), we get
txdx
1 (
rl(115) '°
pi -
Jo
lo
The following equations can be obtained from the second equation of (73):
(116) P{Q)t = —'yo,*cQ^'^^Ux, and
(117) g{cQ)t = g'{cQ)cQt = -a*g'{cQ)]cQ]\x, and
(118) E{cQ)t = c'^+\Q^+^)t = {ß + l)c^+^Q'^Qt = -{ß
Note that we in fact have used (116)-(118) when we conclude that the contribution from boundary terms appearing in (115) is zero. Thus, it follows that
= - / E{cQ)tUxUtxdx + P{Q)tUtxdx- \u\tg{cQ)utxdx Jo Jo Jo
(119) - / ]u\g{cQ)tUtxdx Jo
= {ß + l)a* [ [cQ]0+\ux)^utxdx-fa* [ cQ^+'u^Ut^dx Jo Jo
- / g{cQ)sgn{u)utUtxdx + a* / ]u\g'{cQ)]cQ]''ua:Utxdx.
Jo Jo
For the first term on the right-hand side of (119) we use the splitting
Hence,
(120) l\eQf+'{u,)\txdx < ^ f\eQf+^\u.)Ux-^ei l'E{eQ){ut.)^ dx
Jo 4£i Jo Jo
by application of Cauchy's inequality £ > 0, a6 < (l/4£)a^ -|- eb'^. The last term on the right-hand side of (120) can be adsorbed on the left-hand side of (119), whereas the first is bounded by (109).
Similarly, for the second term on the right-hand side of (119) we have (121) c
Two applications of Cauchy's inequality £ > 0, a6 < (l/4£)a^ 4- eb'^ then give us
o Jo
f
Jo (122) < ^ / '
8£2 Jo Jo
/•I
-f £2 / E{cQ){utx)''dx Jo
<Ki3 + ^ [ [eQ]^^+'\ux)Ux + 62 I
8£2 Jo Jo
where we have used assumption (Al) and (99) to estimate as follows:
/o if
a that is.
a
which clearly holds for a G (0,1) and 7 > /3. As far as the third term on the right-hand side of (119) is concerned, we note that
(123)
Elence, we conclude again from Cauchy's inequality and the fact that ß G (0,1) that
/•I
/ g{cQ)sgn{u)utUtxdx
Jo
<Ku{£3) {ut)^dx + e3 E{cQ){utx)^ dx Jo Jo
- 2 2
Finally, for the fourth term on the right-hand side of (119) we have
V/e can repeat the arguments similar to those of the estimate of (121) leading to (122),
/•I
/ \'u\g' Jo
'{cQ)^[cQf-^{ux)^dx + e, ¡
Jo
^ ' ^ f ^ ^ ¡ ^ dx
fl
(125) -l-£4 / E{cQ){utx)''dx Jo
][/f4 fl
<^ / 5' o£4 Jo/
o fl
-I-£4 / E{cQ){utx)^ dx Jo
^ ¡ f + ^ U ¡ ^ dx,
^ ¡ [cQf+^{ux)Ux-^e4 ¡
o£4 Jo Jo where we have used the splitting
and the properties of g'(-). Clearly, £i, £2, £3, £4 can be chosen such that the associated teirm ¡^ E{cQ){uxt)'^ dx on the right-hand side of (120), (122), (124), and (125) can be incorporated in the corresponding term on the left-hand side of (119).
Conseciuently, in view of (119), (120), (122), (124), and (125) in combination with (109) (Lemma 5.2) (see also (113)), we have
- / {ut)''dx+ /
"Î Jo Jo
dx
for appropriate constants Kie and Kn; that is.
.1 i-t ,1
/ {ut)'^dx+ / / Jo Jo Jo
[I
Following the arguments of [3, 33], we may conclude that by choosing T4(M) > 0 sufficiently small such that for all t G [0, T4] we have
(126) ^ ' K f - ^''
where Ce is independent of M. Hence, for t 6 [0,T], recall that T G (O,ri], and by (95) it follows that Ti < f4. We then have
[[
/ (ut)2d.T-f [ [ o Jo Jo
Application of (a nonlinear version of) Gronwall's inequality then gives
P1 l-t /-I 1
0 Jo Jo ^^,,,niA.M^^?• "~ ^"^17*
where Ce has been defined by (126). D
6. Proof of Lemmas 4.2 and 4.4. Equipped with the results of section 5, we shall return to the proof of Lemmas 4.2 and 4.4. First we give a proof of Lemma 4.4 where we rely on the result of Lemma 5.3. Then we shall derive Lemmas 6.1 and 6.2, the first one provides control over the L^ continuity in time of u, i.e., esti- mate of [[u(-,i) — 'tio(-)[[2> whereas the second one gives control over the quantity
/o io "* dxdt\. Finally, armed with these estimates we can give a proof of Lemma 4.2.
We end this section by Lemma 6.3 which provides more L^ continuity type of esti- mates with respect to time for, respectively, Q, u, and Eux. Then, Lemma 6.5 gives an upper pointwise bound for Eux as well as BV estimates for Eux, u, and Q.
6.1. Proof of Lemma 4.4. Proof. We obtain from the third equation of (73) by integrating over [x, 1] and employing the boundary condition (75)
Q^ \u\g{cQ) E{eQ) E{eQ) \ This implies that
(127) Jo " - - - - 7 0 E{eQ)
:= Ai-\-A2-\
T'hen we estimate as follows:
(128)
.42= i'Q^-^-'c-^-'dx<G:^^+'^ i
Jo Jo
[
Jo
= D2 í <^(x)í Jo
for some constant D2 which is independent of M and where we have used assumption (Al) which implies that
O2 (p[x) "'^ S e S 01 ^0(x) " ' " , I.e., (
together with estimate (93). Here we must aiso assume that a[^7 - j9 -1] > - 1 , that
is,
- > 1 ß-- 4 a '
which follows from assumption (A3). Furthermore, by combining assumption (Al) and (93) we obtain
(129)
Using this estimate together with (97) we can estimate as follows:
:
*)cQ
(130)
fl
<D3 dx i-\ dx
dx)
1 1
x) "\j^ [u\^^dx) , Í
wliere we have used Holder's inequality with p = A and g = | , estimate (87) of Lc;mma 4.1, and the fact that -aßq = -aß^ > - 1 or a/3 < | . As above D3 is a suitable constant that is independent of Ad but may depend on Gi, G2, A, and B.
Finally, we see that
(131)
< Di f \ ( f [ut[^ dy)
dxfi / Jo
for a suitable constant Di = DI{CQ,A,CI) (note dependence on Co since we have used Lemma 5.3) if 1/2 - a{ß + 1) > - 1 ; that is, a{ß -(-1) < | . Hence, in view of (127), (128), (130), and (131) the conclusion is that
lo
for a constant D which is independent of M. In other words, in view of Sobolev's embedding theorem that VK''i([0,1]) - 4 L°°{]0,1]) and Lemma 4.1,
nl nl
< / ]u\dx + / \ux]dx < V2C + D + Jo Jo
[[w[[oo
Clearly, for positive constants D4 = v ^ C + D and D5 = D, we then have (132) l.^D4 + D^M'/^)<\
for a suitable large M. Hence, [['ÍÍ[[OO < \M for a suitable choice of M, and the lemma has been proved. G
6.2. More lemmas under the assumption of (93).
LEMMA 6.1. Under the assumptions of Theorem 3.1 and (93), there exists a positive constant Di = Di{C,Ci,C2,A,B, [[ÍÍO[[^I,T) such that for all t £ [0,T], (133)
/•I
/ ]u{x, t) - uo{x)]'^ dx < Dit.
Jo
Proof. Clearly, from the third equation of (73) we get {u - uo)t + (Q^ - ]u\g{cQ))x = {E{cQ)ux)x-
Multiplying by {u - uo) and integrating over the spatial domain [0,1] together with
application of integration by parts and use of the boundary conditions (75) and the
fact that c/(0) = 0, we arrive at the following form:
(134) = / E{cQ)uxUo,xdx + / Q'^Uxdx- I ]u\g{cQ)uxdx Jo Jo Jo
- f Q'^uo,xdx+ / ]u]g{cQ)uo,xdx.
Jo Jo We can estimate as follows:
Hence, by Cauchy's inequality and use of (93) we get
nl 1 nl nl
/ E{cQ)axUo xdx <— / E{cQ)un dx + e / E{cQ)u^dx Jo ' 4e Jo ' Jo
(135) < -^max(£;(cQ)) / ulxdx + e ( E{cQ)uldx 4e JO ' Jo
/•I
<De + £ / E{cQ)ul dx.
Jo
I'he first t(3rm on the right-hand side of (135) is bounded due to (93) and assumptions (Al) and (A4). Moreover, for the second term on the right-hand side of (134) we have
Hence,
,•1 . . 1 . 1
/ Q''u.,dx<— Q2^[cQ]-(^+i)dx-f£ / E{eQ)uldx (1.36) •^° ^-^o^ -^0^
< D ^ /" </)(.ï)"(î^-'^-^'dx-f £ / E{eQ)aldx.
Jo Jo
For the first term on the right-hand side of (136) we have used an estimate similar to (129) which implies that
Consequently, the first term on the right-hand side of (136) is bounded if a ( | 7 - /3 - 1) > - a > - 1 , i.e., 7 > | / 3 . For the third term on the right-hand side of (134) we have
/ \u\g{eQ)ux dx < / \u - uo\g{eQ)ux dx-\- \ Jo Jo Jo (137)
The first term on the right-hand side of (137) can be treated by the same tech- nique as for Lemma 4.1, the second term can be handled similarly by employing that W(i G L^([C', 1]). The two last terms on the right-hand side of (134) can obviously b(; estimated in the same manner. Using these estimates in combination with (134) implies (133). D
LEMMA 6.2. Under the assumptions of Theorem 3.1 and (93), there exists a positive constant Ei — Ei{M,Di,T) and a positive integer 7n with m > j ^ ^ > 1 such that for allt G [0,T],
(138)
o Jo È i ( l ) { x ) t , 0 < , E < . 2 Proof We can estimate as follows by using Holder's inequality:
I /•' /•'• I r
\ / dtudyds =\ [u{y,t) - uo{y)]dy 'Jo Jo IJo
< / \u{y,t)-uo{y)\dy< xio,x]{y)\u{y,t) - uo{y)\dy Jo • Jo
) , x€ io, ^1,
I
\u{y,t) - uo{y)f
where we use p = m and q = 7n/{7n - 1). Moreover, we have used |[ÍÍO|[OO, [[W|[OO <
from Lemir.a 4.4 as well as the estimate of Lemma 6.1. D
Remark 6.1. In a similar manner we can obtain the estimate /
(139) 1 / / dtudyds 'Jo Jx
Note that the M dependence of Ei (as well as other constants) will be controlled in the following proof of Lemma 4.2 by the "small" time T > 0 wlaere T <Ti and Ti is defined by (95). In particular, suitable small times Ti, T2, and T3 will be introduced in the proof below to eliminate dependence on M.
6.3. Proof of Lemma 4.2. Now we are in position where we can give a proof of Lemma 4.2.
Proof We start with the following equality, which follows directly by combing the second and third equation of (73);
ft ft fX ft
(140) [cQ] + a*/3 / Q~'ds = [cQ]o — a*ß / / utdyds + a*ß / \u\g{cQ)ds.
Jo Jo Jo Jo Here we have combined
and
i + [Q''-\u\g{cQ)]=E{cQ)ux.
First, we observe that from Lemma 6.2 we have
r-
Jo Jo
(141) / / Utdyds < Ei{M)<f>{x)-^t lo Jo
Moreover,
11 m
(142) I / \u\g{cQ) ds < a*M I [cQ] ds < 2a*BG2M(j){x)'^t = È2{M)4>{x)"t.
Uo Jo
Consequently, in view of (140), where we set a*ß = 1 without loss of generality, we can employ (82) from assumption (A2) together with (141) and (142) and estimate as follows;
[cQf + I Q^ds> [C2A]''<^(a;)^" - Ê I ^ ( . T ) ' ^ Î 1 / ' " - È2Î0(a;)°
Jo
since 4){:v)" < (j){x)°'l^ and ^(.x)(™-i)/'" < ^(x)"'' since 771 > 1/(1 - aß); that is, (771— l)/777 > aß as assumed in Lemma 6.2. We have also used that i^/'" > í for í < 1 and 771 > 1. Now we can define
144) i
^ E1+E2
For 0 < Í < f i we get from (143) (145)
To ensure the lower limit of [eQ] from (145) we must determine an upper limit for J,,, Q'^ ds for sufficient small t. For this purpose we set
(146) Z{t)= [ Q^x,s)ds.
Jo
I'hen an inequality equation for Z{t) can be derived. However, first we note that similar to the proof of (143) we get
(147) [eQf + I Q-^ds< {[CiBf + [Ei +
Now we Cii.n define
(148) j , ^ For 0 < t < ^2 we get from (147) (149) [eQ\^
NÍOW we r€:turn to the task of deriving a lower limit for [eQ]. We again follow along the line of [3, 33]. From (147) and (146) we get
tl:iat is,
for K(t) =: {[CiB]^ + [El + È2]t^l'^). Clearly, Z{0) = 0 and then we can deduce, by assuming that 0 < ß < j , that
(150) Z'(t)'^ + KiZ{t) < K2{t) < K2if2), Kl = c - ^ K2 = C;
for t G [0,T2]. That is, we have an ODE inequality of the form (151)
The solution of Z'it^^P + AZ{t) = B can be found by writing this ODE in the form Z'{t) = {B- AZ{t))P.
Introducing U{t) = B - AZ{t) we get
U'{t) = -AU^, U{t = 0) = B.
This gives the solution
u= -At, 1-p
that is.
; - ) = AZ.
From this it follows that the following inequality holds:
(152) AZ<B(I
^ [l + {
In view of (150), (151), and (152) we get the inequality (153)
Z{t) <
1 -
Now we observe that p — 1 = ^^^ > 0; hence
for 7 > ||ö. From this estimate it is clear that we can get the expression
as close to 1 (from above) as desirable for a small enough time interval (that depends on M due to the appearance of A^). Thus, it is also clear that for small enough times, let us say, t £ [0,f3] for T3 = T3(M) > 0, we haye for the right-hand side of (153)
1 (154) ^ ' ^ '
Inserting this in (153) gives us -Z{t) > -ct>{x)-ß Employing this in (145) we get (155) [cQf > -Z{t) In view of (155) and (149) we have
In other words,
(^[^C2])<^(x)" < [cQ] < ( ^ which, clearly, in view of the assumption (79), gives us
Hence, the desired estimate (94) has been obtained. D
6.4. More regularity results in space and time under the assumptions of Theorem 3.1. In the next lemma we obtain ¿^-continuity in time estimates for the quantities {Q,u,E).
LEMMA 6.3. Under the^assumptions^ of Theorem 3.1, we have for 0 < s < t <Ti and appropriate constants Fi, F2, and F3 the following estimates:
(156) / \Q{x,t)-Q{x,s)\''dx<Fi]t-s\\
Jo
/ • I ,
(157) / \u{x,t)-u{x,s)\^dx<F2]t-s\\
Jo
/•I
(158) / \E{cQ)ux{x,t) - E{cQ)ux{x,.s)]''dx < Fa]^ - .s[.
JO
Proof We have, by using the second equation of (73) and Holder's inequality, where we tactically have assumed s < t,
(159)
/ ]Q{x,t)-Q{x,s)]Ux= I \i Q¿
Jo Jo Us
/ 1/
o 'Jo1/
j
= a*]t-s]
y'* -
< Fi{B,
s Jo
'-' dx + x)' dx)
]t - s\ dx {ux)' dx)
Here we have employed Young's inequality. Lemmas 5.2 and 5.3 as well as assumption (A2), and the pointwise estimate of Q given by Lemma 4.2. Next, we consider (157).
We get by Holder's inequality
nl nl nt
/ \u{x,t)-u{x,s)]''dx= / / uç{x,
Jo Jo 'Js
<]t-s\ / uldxd^<C6]t-s\''
Js Jo
dx
where we have used Lemma 5.3 again. Finally, for (158) we estimate as follows by using Young's inequality:
\E{eQ)ux{x,t) - dx =
\f
o 'Js
dx
\j
<\t-s\i \j
= |,_,^'|£(_(,,1K
dx
-t- dx
Ti /.I
by using Lemmas 5.2 and 5.3 and the pointwise upper bound for [eQ] to obtain the last inequality. D
C O R O L L A R Y
6 . 4 . Under the assumptions of Theorem 3 . 1 , we get forO < s < t <
Ti that
(160)
where Fi is the constant from (156).
Proof. First we recall from (68) that ri satisfies the equation 7it + ai^Ux = 0 and that 71 and Q are related by
(161) -=a -
(see (72)), and that Corollary 4.3 implies that (162) en<a*.
Hence, we can follow along the same line as for the estimate (156) and calculate as follows:
/ \7i{x,t) — 7i{x,s)\'^ dx = / / n^{x, Jo Jo 'Js
= I \l
Jo 'Jo
dx
\l
^y Í Í e
< {a*)''Fi{B,
[
dxwhere we have used (161) and (162) and the calculations of the last part of (159). D
LEMMA 6.5. Under the assumptions of Theorem 3.1 and for t G [O,ri], we get
(164) (165) (166)
rU| dx < FQ,
%Q\ dx <
Proof The estimate (165) follows as a byproduct of the proof of Lemma 4.4. The estimate (:L63) follows from (164) since
\\E{cQ)u,]{x,t)\ =
¡ {E{cQ)uy)ydy < ¡
Jo Jo From the momentum equation we have
(:-67) {E{cQ)ux)x = ut- {Q^)x + {\u\g{cQ))x.
The estimate (164) can be obtained by observing
/ \{E{cQ)ux)x\dx< ¡ \ut\dx+ ¡ \{Q^)x\dx+ ¡ \u\xg{cQ)dx Jo Jo Jo Jo
¡ \u\\g{cQ)x\dx J
¡
Jo
¡ \Q''-'Qx
Jo ¡ \'Ux\dx-^ ¡ \u\g'{cQ)[cQ]xdx
Jo Jo
¡
o^ ¡\[cQf)ldx
^P Jo
Here Cauchy's inequality has been used repeatedly, including Lemmas 4.1, 4.2, 5.1, and 5.3, and assumptions (A1)-(A4). Finally, the estimate (166) follows since
/ \Qx\dx< ¡ \[cQ]x-\dx+ ¡
Jo Jo c Jo
^ ¡ ¡
Jo ^P Jo Jo
where we have used Lemmas 4.2 and 5.1 and assumptions (Al)-(A4). G
COROLLARY 6.6. Under the assumptions of Theorem 3.1 and for t G [O,Ti], we get
(168) [ [dxn[dx<F8.
Jo
Proof We have that eQ = en/{a* - en); consequently,
(169) a*-aiJ {a* —
from which we get
1 n^~
i* ')^+^
ß n -
Hence, it follows easily that (170)
\dx + Fs2 I [dxn\dx<F8i [ Jo
o Jo
since ß,a £ (0,1) and by application of Lemma 5.1, Corollary 4.3, and assumptions D
6.5. T h e proof of Theorem 3.1. In order to construct a weak solution to the initial boundary value problem we can directly adopt the standard approach and apply the line method as described in [13, 14, 3, 33, 9]; see also the important references within these papers for more details. Since this will not introduce new elements to what is already found in these papers we leave the details to the readers. Here we just note that having formulated a semidiscrete version of the model (68)-(71), the basic theory of ordinary differential equations then guarantees the local existence of smooth solutions (CJ,71J,'IÍÍ), for i = 0 , . . . , A^ such that
0 < Ci{t) < oo, 0 < ni{t) < oo, \ui{t)\ < oo.
Let [O,T'*] be the maximal time interval on whicli the smooth solution exist. For the analysis below we must show that the solutions are actually locally defined on [O,Ti] where Ti is defined by (95). In particular, it can be ensured that T'' > Ti (see [3, 33]), and (c^, ni, u.¿) are well defined for i = 0 , . . . , A'', for all t G [0, Ti]. Based on the work of sections 4, 5, and 6 we can obtain semidiscrete versions of the various lemmas. By defining appropriate approximate solutions (c''^,7i^,u^)(x, i) and using Helly's theorem, we can prove Theorem 3.1.
REFERENGES
[1] G. S. AvELAR, P. R. RiBEiRO, AND K. SEPEHRNOORI, Deepwater gas kick simulation, J. Pet.
Sei. Eng., 67 (2009), pp. 13-22.
[2] G. E. BRENNEN, Pandarnentals of Multiphase Flow, Gambridge University Press, New York, 2005.
